It wasn't difficult to come out as a lesbian, but it was to come out as a bird watcher

There are paintings, drawings and photographs covering almost every piece of available wall space and a series of sculptures on pieces of genuine Mackintosh furniture. One work catches her attention. It's of a small shoal of anchovies which have been individually encased in glass tubes and then suspended in water in a thick glass box. Reid gives the sculpture a little shake. As she does the glass anchovies all clink together, like delicate brass finger cymbals playing a high-pitched tune.

Reid, who is 62 and has been in her current job since 1999, takes her seat at the round wooden Mackintosh table in the centre of her office and asks: "Have you been to see the Shrigley down in London?".

The Shrigley she's referring to is the David Shrigley Brain Activity exhibition which is currently on at the Hayward Gallery in the Southbank Centre. I know this because I have been, a few weeks before the interview, so I answer yes.

Reid smiles. "Isn't it wonderful? I thought his animations were just brilliant. There was an animation of a man sleeping and it was obviously based on a film of a man asleep. Nothing happened, it was just a man sleeping, and the jaw would go slack and he would be snoring – it was just wonderful.

"It was just real life but you were just mesmerised by it – watching and waiting. I thought they were very, very clever animations."

Then the percussion outside starts up again. The diggers are puncturing deep holes into the earth in preparation for the art school's £50 million Steven Holl-designed campus building. The new glass-covered structure, which according to Reid has "still got about 17, 18 months to go" and is considered to be one of the city's most important architectural projects, will face the iconic Mackintosh building which contains her office.

The development has not been without criticism. One commentator suggested Holl's design "looks like the old building's [the Mackintosh building] robotic cousin" while others have said the glass "skin" of the building will reflect too much light on its historic neighbour.

Reid, though, remains proud of the project – indeed she lists it as one of her greatest achievements during her tenure at the art school. "These are major projects which are impossible to do without disruption," she concedes. "The noise for people in this building and the people who had to move down to Skypark [the business complex in Finnieston where the design school now temporarily resides] and the upsets and the difficulties and the snagging that causes. You just can't do a project like this without breaking eggs."

The new Holl building is just one of the projects going on at Glasgow School of Art right now. At the beginning of this year it was announced the Arts and Humanities Research Council had awarded £122,500 to investigate why Glasgow had produced so many successful contemporary artists. The research, which has just started, will be undertaken by the art school in conjunction with the Centre for Contemporary Arts (CCA).

Does Reid have any personal thoughts on why the city has been so successful? "I think it's a combination of the students, the staff [at the art school] and the city. It's a tremendously strong creative community, so people coming out of this institution will be supported within it. Many students who arrive at Glasgow School of Art from further than the west of Scotland and Glasgow stay. They stay not just because of the art school, but because the artistic community within the city is such a supportive one."

Of course it could be argued (though Reid would never do so herself) that at the heart of the artistic community in Glasgow you'd find the director of Glasgow School of Art. Reid might not be a prize-winning, headline-grabbing artist but there's no doubt her decisions have influence.

As well as her position at the art school, Reid is also on the board of artist residency organisation Cove Park and recently took up the position of chair of the Heritage Lottery Fund in Scotland. She is also the former director of the Scottish Arts Council – a position she held for nine years – and has worked with, among others, Greater London Arts, Ballet Rambert and the Lighthouse Centre for Architecture.

She hasn't always been surrounded by the arts though. Born in Glasgow and educated at Strathclyde University, she admits: "I don't come from an art background – quite the opposite." Yet the arts are her life now.

Recently, she commissioned a group of art school graduates, who go by the name of Arka Design, to redesign the sitting room and hall of her Glasgow home. They did, she says, "a brilliant job" and have created wall panel spaces for her to hang her ever-growing art collection. "There are so many [artworks] at home – I have a pile in the hall."

This art collection, I'm about to discover, is about as close to discussing herself as she can comfortably get. She was recently nominated as a "woman of influence" by the Action For Children Women of Influence Awards, and although she didn't win (acting chair of the Zero Tolerance Trust Dr Lesley Orr did) I wonder how she feels about being described in this way?

There's an audible sigh from the opposite side of the table while she quickly rearranges her body from relaxed to the foetal position. "I'm very, very uneasy about talking about myself at all because I find it awkward."

I can see that, I tell her, from the don't-come-near-me body language. She laughs quietly. She obviously doesn't find these kinds of questions easy to answer. "No, I don't. I never think of myself as being a woman of influence because I don't think of influencing as being a personal ambition. The notion of wanting to be a woman of influence smacks of a kind of personal ambition which I'm not terribly easy with."

The industrial percussion starts up again. The subject changes from women of influence to the issues facing women in positions of power. Has Reid ever had to deal with sexism? "To be honest, I don't think I've ever found it a tremendous issue. Where there have been occasional examples of sexism they've been annoying but dismissible, and just nothing that was sufficient to derail me in the job.

"That's not to say I'm not conscious that it's quite easy for people to be derailed by it, even now. I'm also gay and I think there's some really worrying stuff going on at the moment, in certain areas a growing sense of intolerance."

Reid is confident and open about her sexuality. "I certainly would never ever lie, so if I say something about myself I would have to say that my relationship is with a woman (former actor Cordelia Ditton who is now the director of Voice Business) and not with a man. That makes you conscious all the time that one's sexuality is different from the majority within society.

"I think it's maybe because I am very confident about my sexuality that I've never experienced any issue with it, which is pretty amazing. But I have worked in a privileged environment in the arts and I think the story might be different if I was in engineering, or construction or an area like that."

Sitting in this Mackintosh-designed office, surrounded by art, it's hard to argue with the notion that the arts hasn't been a privileged environment for Reid. Does she ever think she'll leave? "I certainly don't want to go on forever – I don't think it would be good for me and I don't think it would be good for the school. My horizon at the moment is to finish the [Steven Holl] building. After that I don't really know."

She says she would like some more time to develop her interests outside art. She's a fan of fly fishing (the Scottish Arts Council bought her a complete kit when she left) and would like to travel more.

But there's something else, something she has been hiding. A little secret. "We were talking earlier about my sexuality and the interesting thing is that I never found it difficult to come out as a lesbian, but I have found it difficult to come out as a bird watcher."

Staff at Glasgow School of Art take note, when she decides it's time to step down as director, a pair of binoculars might be a suitable leaving gift.